CHGEQZ(l) ) CHGEQZ(l)NAME
CHGEQZ - implement a single-shift version of the QZ method for finding
the generalized eigenvalues w(i)=ALPHA(i)/BETA(i) of the equation det(
A - w(i) B ) = 0 If JOB='S', then the pair (A,B) is simultaneously
reduced to Schur form (i.e., A and B are both upper triangular) by
applying one unitary tranformation (usually called Q) on the left and
another (usually called Z) on the right
SYNOPSIS
SUBROUTINE CHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB,
ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, RWORK,
INFO )
CHARACTER COMPQ, COMPZ, JOB
INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, LWORK, N
REAL RWORK( * )
COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), Q(
LDQ, * ), WORK( * ), Z( LDZ, * )
PURPOSE
CHGEQZ implements a single-shift version of the QZ method for finding
the generalized eigenvalues w(i)=ALPHA(i)/BETA(i) of the equation det(
A - w(i) B ) = 0 If JOB='S', then the pair (A,B) is simultaneously
reduced to Schur form (i.e., A and B are both upper triangular) by
applying one unitary tranformation (usually called Q) on the left and
another (usually called Z) on the right. The diagonal elements of A are
then ALPHA(1),...,ALPHA(N), and of B are BETA(1),...,BETA(N).
If JOB='S' and COMPQ and COMPZ are 'V' or 'I', then the unitary trans‐
formations used to reduce (A,B) are accumulated into the arrays Q and Z
s.t.:
Q(in)A(in)Z(in)* = Q(out)A(out)Z(out)*
Q(in)B(in)Z(in)* = Q(out)B(out)Z(out)*
Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix
Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
pp. 241--256.
ARGUMENTS
JOB (input) CHARACTER*1
= 'E': compute only ALPHA and BETA. A and B will not necessar‐
ily be put into generalized Schur form. = 'S': put A and B
into generalized Schur form, as well as computing ALPHA and
BETA.
COMPQ (input) CHARACTER*1
= 'N': do not modify Q.
= 'V': multiply the array Q on the right by the conjugate
transpose of the unitary tranformation that is applied to the
left side of A and B to reduce them to Schur form. = 'I': like
COMPQ='V', except that Q will be initialized to the identity
first.
COMPZ (input) CHARACTER*1
= 'N': do not modify Z.
= 'V': multiply the array Z on the right by the unitary tran‐
formation that is applied to the right side of A and B to
reduce them to Schur form. = 'I': like COMPZ='V', except that
Z will be initialized to the identity first.
N (input) INTEGER
The order of the matrices A, B, Q, and Z. N >= 0.
ILO (input) INTEGER
IHI (input) INTEGER It is assumed that A is already upper
triangular in rows and columns 1:ILO-1 and IHI+1:N. 1 <= ILO
<= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
A (input/output) COMPLEX array, dimension (LDA, N)
On entry, the N-by-N upper Hessenberg matrix A. Elements below
the subdiagonal must be zero. If JOB='S', then on exit A and B
will have been simultaneously reduced to upper triangular form.
If JOB='E', then on exit A will have been destroyed.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max( 1, N ).
B (input/output) COMPLEX array, dimension (LDB, N)
On entry, the N-by-N upper triangular matrix B. Elements below
the diagonal must be zero. If JOB='S', then on exit A and B
will have been simultaneously reduced to upper triangular form.
If JOB='E', then on exit B will have been destroyed.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max( 1, N ).
ALPHA (output) COMPLEX array, dimension (N)
The diagonal elements of A when the pair (A,B) has been reduced
to Schur form. ALPHA(i)/BETA(i) i=1,...,N are the generalized
eigenvalues.
BETA (output) COMPLEX array, dimension (N)
The diagonal elements of B when the pair (A,B) has been reduced
to Schur form. ALPHA(i)/BETA(i) i=1,...,N are the generalized
eigenvalues. A and B are normalized so that
BETA(1),...,BETA(N) are non-negative real numbers.
Q (input/output) COMPLEX array, dimension (LDQ, N)
If COMPQ='N', then Q will not be referenced. If COMPQ='V' or
'I', then the conjugate transpose of the unitary transforma‐
tions which are applied to A and B on the left will be applied
to the array Q on the right.
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= 1. If COMPQ='V'
or 'I', then LDQ >= N.
Z (input/output) COMPLEX array, dimension (LDZ, N)
If COMPZ='N', then Z will not be referenced. If COMPZ='V' or
'I', then the unitary transformations which are applied to A
and B on the right will be applied to the array Z on the right.
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= 1. If COMPZ='V'
or 'I', then LDZ >= N.
WORK (workspace/output) COMPLEX array, dimension (LWORK)
On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= max(1,N).
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
RWORK (workspace) REAL array, dimension (N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
= 1,...,N: the QZ iteration did not converge. (A,B) is not in
Schur form, but ALPHA(i) and BETA(i), i=INFO+1,...,N should be
correct. = N+1,...,2*N: the shift calculation failed. (A,B)
is not in Schur form, but ALPHA(i) and BETA(i), i=INFO-
N+1,...,N should be correct. > 2*N: various "impossible"
errors.
FURTHER DETAILS
We assume that complex ABS works as long as its value is less than
overflow.
LAPACK version 3.0 15 June 2000 CHGEQZ(l)
